Ultra-violet finiteness of supersymmetric non-linear sigma models

Nuclear Physics B260 (1985) 182-202 © North-Holland Publishing Company

ULTRA-VIOLET FINITENESS OF SUPERSYMMETRIC SIGMA MODELS

NON-LINEAR

C.M. HULL

The Institute for Advanced Study, Princeton, New Jersey 08540 and Institute for Theoretical Physics, UCSB Santa Barbara, California 93106 and Department of Mathematics, MIT, Cambridge, Massachusetts 02139 Received 6 March 1985 It is shown that N = 4 supersymmetric non-linear sigma models in two spacetime dimensions are ultra-violet finite to all orders in perturbation theory.

1. Introduction The b o s o n i c n o n - l i n e a r s i g m a m o d e l in two s p a c e t i m e d i m e n s i o n s has the action

S = f d2xg,j(q~)a,~'(x)aU~J(x),

(1.1)

where the real scalar fields q~(x), i = 1 , . . . , d, can be r e g a r d e d as the c o o r d i n a t e s o f some d - d i m e n s i o n a l m a n i f o l d , M, with positive-definite metric g0(q~). The most c o m m o n l y c o n s i d e r e d cases are those in w h i c h M is a coset space. T h e a c t i o n (1.1) is i n v a r i a n t u n d e r g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s o f the internal m a n i f o l d M. W e shall a s s u m e M is i r r e d u c i b l e , i.e. n o t a direct p r o d u c t M1 x M2 - this w o u l d c o r r e s p o n d to a m o d e l with two sets o f fields that do not interact with each o t h e r a n d so e a c h c o u l d be c o n s i d e r e d s e p a r a t e l y . The n o n - l i n e a r s i g m a m o d e l is not r e n o r m a l i z a b l e in the usual sense as i n d e p e n d e n t d i v e r g e n c e s o c c u r at every o r d e r in p e r t u r b a t i o n theory. F r i e d a n has shown [ 1 ] that, at a r b i t r a r y l o o p order, there is an on-shell d i v e r g e n c e c o r r e s p o n d i n g to a g e n e r a l - c o o r d i n a t e - i n v a r i a n t c o u n t e r t e r m o f the f o r m

Sct= f d2x To(g~)O,(Pia~o j ,

(1.2)

where T~j(q~) is a s y m m e t r i c t e n s o r c o n s t r u c t e d a l g e b r a i c a l l y from the curvature Rokt(~) o f M a n d its c o v a r i a n t derivatives. The c o u n t e r t e r m (1.2) is n o t p r o p o r t i o n a l to (1.1), as, in general, T~j is not p r o p o r t i o n a l to go. This can, however, be i n t e r p r e t e d as a r e n o r m a l i z a t i o n o f the c o m p o n e n t s o f the metric t e n s o r [ 1], with c o r r e s p o n d i n g 182

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renormalization-group equation O I.ZO~ go = -3o(gk,) ,

(1.3)

where ~ is the renormalization scale parameter. Explicit calculations give the fi-function to two-loop order [1] 1 K~ D klm 30 -~ gk, ) = Ro +~1.~ .... ikt,,,,j +O(fi2)

(1.4)

(R o is the Ricci t e n s o r Rkiki.) Thus the geometry of M is changed by quantum corrections. The counterterm (1.2) is universal in that it has the same expression in terms of the curvature and its covariant derivatives for all spaces M and dimensions d. The N = 1 supersymmetric extension of (1.1) [2] has the superspace action S

f dZx

3

d 2 O g o ( ~ ) D a @ i DSI~,"

(1.5)

where O ~ ( a = 1, 2) are anti-commuting two-component Majorana spinor variables and the supercovariant derivative is

D,~ -

3 -

i(y'O)~O,.

(1.6)

The manifold coordinates are now taken to be superfields @'(x', 0 " ) whose component expansion is qS'(x, O) = ~ ' ( x ) + 6'~Ok(x)+lSc'O~,F'(x)

(1.7)

where ~' are real scalar fields, ~'~ are two component fermion fields satisfying the Majorana condition q,2 = (4,~) * and Fi(x) are real auxiliary fields. For a review of these models, see [3]. In discussing the geometry of these models, the superfields q~ and bosonic coordinates ~ can be used interchangeably - g0(q ~) is completely determined by g0(~), and conversely. The on-shell L-loop counterterm for the supersymmetric theory takes the form [4, 5] sLt = f d 2 x d2t~ --0 T(L)(, ~ ) D a q~iD,fl ~"

(1.8)

with --q T (L) again a symmetric tensor given by a product of curvature tensors and their covariant derivatives, with indices contracted using the metric. As in the bosonic case, the divergences correspond to a (generalized) renormalization of the metric, which again satisfies renormalization group equations (1.3). Explicit calculations

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give the one- and two-loop counterterms for the supersymmetric model as [4, 5]

1 i.i T!3) = 41re R ° ' (1.9)

1

T~2) =

16( 2 7re)z (VkV kRo + 2[V i, V k ] R k ) ,

where ~7i denotes covariant differentiation with the Christottel connection for M. From (1.9), it follows that the supersymmetric theory will be finite at one and two loops if the internal space is Ricci-flat (R 0 = 0) while if it is a locally symmetric space (~7~Rjk~,,= 0) the two-loop divergences will cancel. Alvarez-Gaum6 and Freedman have conjectured [4] that the models defined on Ricci-flat spaces are finite to all orders, while those defined on locally symmetric spaces have only a one-loop divergence. (This has been checked at the 3-loop level [6].) In [7], they argued that the N = 4 supersymmetric model (in which M is necessarily Ricci-flat), in the special case that the dimension d of M is four, is finite to all orders. The purpose of this paper is to show the finiteness of all N = 4 models, irrespective of the dimension d of M. It is of great interest to ascertain which 2 - D sigma models have zero /3-function, as these are likely to play a key role in superstring theory [8] - the manifold M of such models is a candidate for a vacuum for the string [9]. In [4, 7], an important role was played by the universality of the counterterm (1.8) - the fact that its expression in terms of curvatures and their derivatives is independent of the manifold M. If the model has N = 2 supersymmetry, the internal space must be Kiihler [10, 7]. This severely restricts the possible counterterms [4] as it implies that the L-loop quantum correction to the metric, T~)(g), must be a K~ihler metric on M whenever go is. If the model has N - - 4 supersymmetry, the geometry of the internal space is even more restricted [7] (it is hyper-K/ihler) and this will be used here to put further constraints on the counterterms.

2. Extended supersymmetry and complex geometry The action (1.5) is manifestly N = 1 supersymmetric as it is in superfield form. We now wish to consider the possibility of it admitting further supersymmetries, following [7]. This leads to a number of restrictions on the geometry of M; for further discussion of the complex geometry involved, see [11, 12, 13] and appendix C of [14]. The most general form an infinitesimal extra supersymmetry transformation can take, from dimensional considerations and subject to Lorentz and parity invariance, is, in N = 1 superfield form, ~4~' = J~(~)e~D~4~

(2.1)

with J/;(cib) some tensor field of type (I, I) on M and e ~ the constant Majorana spinor parameterizing the transformation. (For a discussion of models in which the requirements of parity-invariance is dropped, see [14].)

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For the commutator of two supersymmetry transformations of the form (2.1) to close on a translation when the field equations derived from (1.5) are satisfied, it is necessary and sufficient that J~j satisfy i J i kJ kj = -¢~j,

(2.2)

N~jk ~ J/OE,Jj] k - Jj'a[,J~3k : O.

(2.3)

Then (2.2) implies that JJj is an almost complex structure while the vanishing of the Nijenhuis tensor N~jk (2.3) is the condition for J~ to be integrable so that J~ is a complex structure (i.e. an integrable almost complex structure) [12, 13]. The integrability of J~ implies that complex holomorphic coordinates can be chosen (O i= (z ~, :~) with a , / 3 = 1 , . . . , n; i , j = 1. . . . . d = 2n and (z~) * = ~ ) such that the complex structure is constant, OkJji = O, j~=(Jot~

j0t~ ) =(i60~s

_i~t3).

(2.4)

Note that it is necessary that J~ have an equal number of holomorphic eigenvectors O/Oz ~ and anti-holomorphic eigenvectors 0/0~ ~ for M to be a real manifold, so that

M is even-dimensional. This form of the complex structure is preserved under holomorphic changes of coordinates, z-* z'(z). For the action (1.5) to be invariant under (2.1) it is necessary and sufficient that the space be hermitian, i.e. Jo =- g,kJ~ = -Jj,

(2.5)

and that the K i i h l e r f o r m Jij be closed oE;,J,j3 = 0

(2.6)

so that the manifold is Kiihler [10]. The hermiticity condition (2.5) implies that, in a holomorphic coordinate system, the metric has the form go =

(0

g~

while (2.6) implies that the metric can be obtained from a single real function K ( z ~, ~ ) known as the K/ihler potential, 02 g ~ = c3z'~ cg~fi K .

(2.8)

Eq. (2.6) is equivalent to the covariant constancy of the complex structure [11, 13] vjjkl

=

0.

(2.9)

The (first) integrability condition for (2.9) is that the curvature, regarded as a two-form taking values in the Lie algebra of S O ( d ) , / ~ _=_ 1~R ijk; dq~k ^ d~o;, commutes with the complex structure, [/~, J] =0, so that in fact / ~ takes values in the U(n) subgroup of SO(2n)(d = 2n). By considering higher order integrability conditions

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for (2.9) one finds that the Lie algebra generated by the curvature and its covariant derivatives, i.e. the algebra of the (restricted) holonomy group, is necessarily U(n) or a subgroup thereof, [12]. The conditions (2.2), (2.3), (2.5), (2.9) given above for a manifold to be K~hler are not independent. Indeed, if (2.2), (2.5) and (2.9) hold, then the integrability condition (2.3) follows. Further, if, as we are assuming, the manifold is irreducible, then (2.5) follows from (2.2) and (2.9). To see this, suppose that J~j = Si; + A;j with Sij = Sji, A o = - A j i so that, from (2.9), •iSjk = O. If the d real eigenvalues of S / h a v e degeneracies dl, d2, . • . , d , ( dl + d2 + " • • + dr = d ) the integrability conditions for ViSj k = 0 are that the (non-restricted) holonomy group is in O ( d l ) × O ( d 2 ) x . . . × O(dr) [12]. This, however, implies that the manifold M is a direct product of r spaces Ms, ( a -- 1, 2 , . . . , r), the dimension of Ms being d~, with a product metric [12]. This contradicts our assumption of irreducibility unless S~j = a(3~j for some constant a. Then Jij = ag~j+ A~ and j i = ad. However, (2.2) implies that J~ has eigenvalues ±i, which must occur in complex conjugate pairs, i.e. n eigenvalues of i and n of - i . Then Jgi = 0 and so ~ = 0 , J ~ j = A o and the space is hermitian, (2.5). Thus (2.2), (2.9) are sufficient conditions for an irreducible space to be Kiihler and the only identities satisfied by the curvature tensor for such spaces but not for general manifolds are the integrability conditions for (2.9). These imply that the only non-vanishing components of the curvature tensor are [ 11, 12, 13] R ~ ~ , which satisfy the cyclic identity R~t3~g = R ~ ~t3~

(2.10)

and those related to these by symmetry or complex conjugation. We shall need the result that the Ricci tensor for a K/ihler manifold is given by R ~ = a~0~ log det [gvg] •

(2.11)

For N = 4 supersymmetric models, M has dimension d = 4m and there are three extra supersymmetries 6 q)i = J ( X ) i ) e ( ~ ) ' ~ D ~ J ,

(2.12)

x = 1, 2, 3 corresponding to three linearly independent complex structures j(x)~, each of which satisfies (2.2), (2.3), (2.5), (2.6). The extended supersymmetry algebra implies that the complex structures satisfy the quaternion (SU(2)) relations j(x)ikj(y)k

= __6xy~ij +

e -. . . .J. (z)ij .

(2.13)

Then M is hyper-Kiihler. There is a holomorphic coordinate system for each complex structure. However, in the coordinates in which j(3), say, is constant (2.4), the other two complex structures, jo) and j(2), will not, in general, be constant. From (2.9), the SO(4m)valued curvature 2 - f o r m / ~ and its covariant derivatives must commute with each of J(1), j(2), j(3) so that the holonomy group is in Sp(m). This is, in fact, a sufficient

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condition for the manifold to be hyper-K~ihler. I f a 2n-dimensional Kiihler manifold has holonomy SU(n), or a subgroup thereof, then in a holomorphic coordinate system, R ~ is traceless, R ~ = 0, which gives from (2.10) R~

= R~g

= R ~ g = O.

(2.14)

Then manifolds with holonomy SU(n) are Ricci-flat K~ihler spaces [12]. As Sp(m) is a subgroup of SU(2m), it follows that all hyper-Kfihler spaces are Ricci-flat, although the converse is only true when d = 4. For a general 4-dimensional manifold, the holonomy group is SO(4) -~ SU(2) x SU(2) = Sp(1) x Sp(1). A d = 4 hyper-K&ihler space has holonomy group Sp(1) so that the curvature must be either self-dual or anti-self-dual Rijkl = :t-~eom"Rm,k~

(2.15)

and M is a Ricci-flat (anti-)self-dual gravitational instanton.

3. Symmetries of on-shell counterterms The arguments of Alvarez-Gaum~ and Freedman [4, 5, 7] require the assumption that the on-shell counterterms are invariant under the symmetries of the classical action (supersymmetry and reparameterizations of the internal manifold) and this will also be assumed here. (Explicit calculations show that the counter-terms indeed have these symmetries up to three loops [6].) This presumably requires the existence of infra-red and ultra-violet regularization prescriptions that are compatible with these symmetries. Such an infra-red regularization can be given by adding supersymmetric mass-terms and interactions to the theory [5, 15]. Possible candidates for ultra-violet regularization schemes include higher derivative regularization and dimensional reduction [16], which was used in [5]. In the background field method, each field is divided into a "background" field and a " q u a n t u m " field and the on-shell counterterms are invariant under a classical symmetry if the " q u a n t u m " fields transform linearly under that symmetry (assuming the existence of a suitable regularization), but the counterterms need not be symmetric for non-linear invariances [17]. For the sigma-model, the on-shell counterterms are invariant under diffeomorphisms of M as the normal coordinate expansion gives " q u a n t u m " fields with a linear transformation law [5]. The counterterms will also be N = 1 supersymmetric as superfield methods give a manifestly supersymmetric perturbation theory. For N = 2 extended supersymmetry, the extra supersymmetry has the form (2.1), which is non-linear in a general coordinate system. However, in a coordinate system in which the complex structure takes the constant form (2.4) the second supersymmetry becomes linear and, as the on-shell effective action is independent

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o f coordinate choice, the on-shell counterterms should be N = 2 supersymmetric. (In such a coordinate system, N = 2 superfields can be used to give a manifestly N = 2 supersymmetric perturbation theory.) For N = 4 supersymmetry, there is no coordinate system in which all four supersymmetries are linearly realised at the same time, but one can choose coordinates in which any one of the three extra supersymmetries is linear, and so is a symmetry o f the on-shell counterterms. As the effective action is coordinate independent, it should be invariant u n d e r each of the extra supersymmetries, that is, should be N -- 4 supersymmetric. From sect. 2, the conditions for the L-loop correction to the metric --0 T (L) to be N = 4 supersymmetric are that it be hermitian with respect to each of the complex structures j(x)~, x = 1, 2, 3 (2.5): j!?,.L) ,j = T(L)1(x)k --~k J j

= _l!X,L)

_j,

,

X = 1, 2, 3

(3.1)

and that each t ~'L) be curl-free (2.6) v

0

r(x,L) _ rkJ0] --0,

X=1,2,3.

(3.2)

Then (3.1) and (3.2) imply that -T(L) 0 is a hyper-KShler metric on M, although it need not be positive definite. However, go =--go + AT~L)

(3.3)

will be a positive definite metric for all values o f the real constant h such that I h [ < )to

(3.4)

for some ho. (We will need a definite signature in order to be able to define the inverse metric, Christoffel connection and curvature.) As go is hyper-K~ihler, it satisfies (2.5) and (2.6) for each complex structure J(~)~ which, in conjunction with (3.1) and (3.2), implies that g0 satisfies gikJ (x)k = --gjkJ(X)k i

(3.5)

O[,(a(~)k~,lk) = 0 .

(3.6)

Thus go is a hyper-K~ihler metric on M that is positive definite for all h such that I,~l < ,~o. In particular, this implies that go is Ricci-flat. For the non-linear equation Rij[g] = 0

(3.7)

to have a linear space o f solutions of the form (3.3) greatly constrains the tensor T~L). This, together with the fact that T- 0(L) must be constructed from curvatures and their covariant derivatives, will be seen in the next section to force the counterterm T!L) to vanish. u

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189

4. Finiteness of the N = 4 model

For any N = 4 model, choosing complex coordinates (z ~, ~?#) in which one of the complex structures, j(3)~ say, has the constant form (2.4), the hermiticity (3.1) of the L-loop correction to the metric, _,/T !~), with respect to j(3)~ implies that it takes the form (cf. (2.7)) T(L)

(

0 -

i

0

T(L)\

T~L)

";t~)

-,k ~

=

(4.1)

(4.2)

T g

and similarly for gu. Using the fact that the Ricci tensor R ~ [ g ] of gu vanishes and the identity (2.11), the Ricci tensor R~fi(g) of gu can be written (for IA I< ho) [18] R~fi[~] = a~afi Tr log ( g ~ ) = O~cgfiTr log [(6r a + ATv(L)8)ga~] = O~0~ Tr log (6v a + AT~(L)a ) + R , ~ [ g ]

(- x ) °

= n=l

c9~c)fi Tr ( T(L~"),

(4.3)

n

where Tr

( T (L)n) ~ T ( L ) a ~ T ( L ) . . . . . --ctt --~2

T (L)°q cG, •

(4.4)

Then as (4.5)

g~#[g] = 0

for all Ih] < ho, it follows that a~a~

Tr ( T (L)") = 0

(4.6)

for all L, n. Defining the real scalar An, L ~ T(L)i2 T(L)i3 . . . T(L)il I1

~12

in

= T(L)~2T(L)%... °¢1 °~2

T ( L ) % + T(L)a~T(L)a3 . . T(L) a, °~n &l ~ oe2 * &.

(4.7)

(4.6) implies a

0 (4.8)

oz ~ o--~ A,~r = O,

so that A L= F(z~)+

(4.9)

p(~a)

for some function F. The real scalar A,,L has this form in

any

h o l o m o r p h i c coordinate

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C.M. Hull/Supersymmetric non-linear o" models

system - under a holomorphic change of coordinates z ~ z'(z), the function F also changes, F ( z ) ~ F ( z ( z ' ) ) , but remains an analytic function. Repeating this argument in coordinates (w ~, ~ ) in which one of the other complex structures, J(~)~ say, has the constant form (2.4), one obtains for the scalar function (4.7) (4.10)

A,,L = G ( w '~) + G ( ~,s)

for some function G. However, the complex coordinates (w, ~) are necessarily related non-holomorphically to (z, 5) [3] w ° = w°(z, 5),

~

c3W~

= ~(z,

5)

cOW~ Off~ ~ 0

# 0

(4.11)

Then (4.10) becomes, in (z, 2) coordinates (as Zl,,t transforms as a scalar, not a density, under coordinate transformations) a.,~ = G ( w( z,

~))+ d(~(z, 5)),

(4.12)

so that 8 0 Oz ~ 05 ~ A,,t_ # 0

(4.13)

in a general holomorphic coordinate system (z, ~) adapted to j(3) unless G is a constant function. Then (4.13) contradicts (4.8) unless G, and hence ZI,,L, is constant. It follows that det ( 6 j + hT~ L)j) = det (S j + hT~ L)j) = exp

a,,L ---f(,~)

(4.14)

n=l

is a polynomial in h with constant coefficients (i.e. independent of z, ~), where T~) U

~-

V~k~l IT(L) rf lkl

(4.15)

T ~t) in an orthonormal frame with vielbein Vi 5. (Carats are the components of _~ denote tangent space indices i ' , j . . . ) . In coordinates (z, ~) adapted to ji3,j, the complex structure j(l)~ must take the off-diagonal form

where ~gg2~ = -6%

(4.17)

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191

so that (3.1) implies that T~LI~ _/2~ / ~

~L~

(4.18)

and T r ( T IL~n)z [ T r ( T ( L ) " ) ] * -- -±2 A

n,L

(4.19)

is a real constant. We also have that h(A) ~ det (3~ e + A T ~ L~) = h(A)* = ~ff(A),

(4.20)

a,h(a ) = 0 . From (4.14), the eigenvalues o f the symmetric matrix T! L)J are constant over the manifold and the o r t h o n o r m a l frames can be chosen such that T!L)J(z, ~) is a constant diagonal matrix. This, together with equations (3.1), (3.2), places strong restrictions on the tensor _,j T !~) . It can be shown that this implies that the tensor T-i;(L) must be proportional to the metric, -T0( L ) -- Kgij, with K some constant, or otherwise the spin connections wo k would satisfy certain identities that it seems cannot be satisfied on general hyper-K/ihler manifolds. However, as this does not eliminate the possibility o f the renormalized metric being proportional to the original metric, and so does not quite lead to the finiteness result desired, it is easier to argue as follows. (In a p p e n d i x B, it will be shown that if M is c o m p a c t [19], or if certain assumptions are made concerning the b o u n d a r y conditions satisfied by the curvature and metric for n o n - c o m p a c t M, (4.20) implies that the q u a n t u m corrections to the metric are intimately connected with the t o p o l o g y o f M, placing great restrictions on the allowed counterterms. In particular, for d ---4, the models defined on T a u b N U T or E g u c h i - H a n s e n spaces are finite to all orders, while for K3, T~ ~0 L) can only be proportional to &j). U n d e r a constant rescaling o f the metric

gis-->A

lgij

(4.21)

the L-loop counterterm must rescale as [4] T{L) ~ A L 1--,sTqt'),

(4.22)

so that, from (4.7), (4.16), (4.17)

A., t -->A

nLAn, L .

(4.23)

Then A.,L is a scalar constructed from curvatures and their covariant derivatives that scales as (4.23) and is a constant over the manifold. If d.,r is made from (5 curvature tensors and 2 D covariant derivatives, then (4.21), (4.23) and the requirement that there be as m a n y covariant as contravariant indices in the expression for A imply that C + D = nL (see appendix B), so that the scalar can never consist entirely o f metrics. If A.,L is not to vanish identically for all N = 4 models, there must be a universal scalar that is constructed from the Weyl curvature and its covariant derivatives that is constant for all hyper-Kfihler manifolds and is non-zero

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for s o m e hyper-K~ihler space. Such scalars o n l y exist on special spaces (e.g. spaces o f c o n s t a n t curvature) a n d w o u l d not be e x p e c t e d to exist on a g e n e r a l hyper-K~ihler space. W e i m m e d i a t e l y have that A,,L m u s t v a n i s h on any s p a c e that such scalars vanish. In a p p e n d i x A, it will be shown that no such n o n - z e r o c o n s t a n t scalars exist on h y p e r - K S h l e r m a n i f o l d s , so that A,,L = 0 for all n, L a n d for all N = 4 models. In p a r t i c u l a r , this rules out r e n o r m a l i z a t i o n s o f the form _T, j!~-) = K g ~ with c o n s t a n t K, as K m u s t scale as K--' ALE u n d e r (4.23) a n d so m u s t be a s c a l a r c o n s t r u c t e d from C c u r v a t u r e tensors a n d 2 D c o v a r i a n t derivatives, with C + D = L, that is c o n s t a n t o n all h y p e r - K / i h l e r m a n i f o l d s a n d so m u s t be zero. (The p o s s i b i l i t y o f such scalars existing constitutes a l o o p - h o l e in the finiteness p r o o f o f [7]. T h e r e it was a r g u e d that when the real d i m e n s i o n o f M is four, any c o r r e c t i o n to the metric m u s t be a zero m o d e o f the L i c h n e r o w i c z l a p l a c i a n a n d so must have negative c o n f o r m a l weight, c o n t r a d i c t i n g (4.17) unless the c o u n t e r t e r m vanishes. H o w e v e r , m u l t i p l y i n g such a z e r o - m o d e b y a c o n s t a n t s c a l a r c o n s t r u c t e d from the c u r v a t u r e a n d its derivatives, if such existed, w o u l d lead to a c o u n t e r t e r m with positive c o n f o r m a l weight that c o u l d not be r u l e d out by this a r g u m e n t . ) W i t h zI,.L = 0 for all n > 0 a n d for all h y p e r - K / i h l e r spaces, however, (4.14) gives f ( A ) = 1, so that all the eigenvalues o f the real s y m m e t r i c matrix T! L)~ are zero a n d the c o u n t e r t e r m vanishes identically, T ~. ) = _~ T' i_~-_0 .

(4.24)

This is true for all values o f L, so that we c o n c l u d e that all the on-shell c o u n t e r t e r m s o f the N = 1 s u p e r s y m m e t r i c n o n - l i n e a r s i g m a m o d e l must vanish i d e n t i c a l l y w h e n e v e r the m a n i f o l d M is hyper-K~ihler a n d the m o d e l has three extra s u p e r s y m metrics, s u b j e c t to the a s s u m p t i o n that the N = 4 s u p e r s y m m e t r y is u n b r o k e n b y the q u a n t u m theory. T h a t is, the N = 4 m o d e l is finite, The v a n i s h i n g o f the /3-function for the N = 4 m o d e l p l a c e s c o n s i d e r a b l e constraints on the c o u n t e r t e r m s o f the N = 1 theory. I n d e e d , these restrictions force the c o u n t e r t e r m s to v a n i s h for all N = 1 m o d e l s defined on Ricci-flat m a n i f o l d s ; this will be d i s c u s s e d e l s e w h e r e [20]. It w o u l d also be o f interest to c o n s i d e r the ultra-violet structure o f the s u p e r s y m m e t r i c s i g m a m o d e l with W e s s - Z u m i n o term [14, 21]. T h e s e are d e f i n e d on m a n i f o l d s with t o r s i o n a n d it seems r e a s o n a b l e to c o n j e c t u r e t h a t they will be finite w h e n e v e r the g e n e r a l i z e d Ricci t e n s o r ( c o n s t r u c t e d from the c u r v a t u r e o f the c o n n e c t i o n with t o r s i o n ) vanishes. I w o u l d like to t h a n k D.Z. F r e e d m a n , D. Page, C. Pope, M. Ro~ek a n d K. Stelle for h e l p f u l discussions. The a u t h o r was s u p p o r t e d by an S E R C fellowship. L. A l v a r e z - G a u m d a n d P. G i n s p a r g have i n d e p e n d e n t l y s h o w n that N = 4 m o d e l s defined on c o m p a c t or a s y m p t o t i c a l l y flat spaces are finite, arguing that this m a k e s it highly p l a u s i b l e that m o d e l s defined on general h y p e r - K ~ h l e r m a n i f o l d s are also finite [ H a r v a r d p r e p r i n t s (1985)]. I w o u l d like to t h a n k P. G i n s p a r g for d r a w i n g my a t t e n t i o n to reference [19].

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193

Appendix A In this a p p e n d i x , it will be s h o w n that there can be no scalars c o n s t r u c t e d from curvatures a n d their c o v a r i a n t derivatives that are c o n s t a n t over all h y p e r - K S h l e r m a n i f o l d s b u t n o n - v a n i s h i n g on at least s o m e h y p e r - K ~ h l e r spaces. A K S h l e r m e t r i c is d e t e r m i n e d (locally) by c h o o s i n g a real f u n c t i o n o f c o m p l e x v a r i a b l e s (2.8). H o w e v e r , it is not k n o w n in g e n e r a l h o w to c h o o s e a K~ihler p o t e n t i a l so that the c o r r e s p o n d i n g metric is h y p e r - K S h l e r , a l t h o u g h this is k n o w n for certain classes o f s p a c e s with isometries. F o r e x a m p l e , L i n d s t r o m a n d Ro~ek have f o u n d explicitly the metrics o f all 4 m - d i m e n s i o n a l h y p e r - K S h l e r spaces with m c o m m u t i n g h o l o m o r p h i c Killing vectors (i.e. with i s o m e t r y g r o u p U(1) m) [22] (see also [3]). W h e n m = l, these i n c l u d e the m u l t i - E g u c h i - H a n s e n a n d m u l t i - T a u b - N U T gravitational i n s t a n t o n s [26], while for m > 1 these i n c l u d e the C a l a b i metrics [23]. With this explicit k n o w l e d g e o f the metric it can be s h o w n that any c o n s t a n t s c a l a r c o n s t r u c t e d from the curvatures a n d c o n n e c t i o n s m u s t v a n i s h for the Linds t r o m - R o ~ e k spaces. T h e n the c o u n t e r t e r m s m u s t vanish on these spaces, while from a p p e n d i x B the c o u n t e r t e r m s are greatly restricted on c o m p a c t a n d a s y m p t o t i cally flat h y p e r - K ~ h l e r m a n i f o l d s . A l t h o u g h intuitively unlikely, it r e m a i n s a logical p o s s i b i l i t y that the c o u n t e r t e r m s a n d c o n s t a n t s An.L vanish on these spaces but are n o n - z e r o for s o m e o t h e r h y p e r - K S h l e r m a n i f o l d s . (As these spaces have s o m e special p r o p e r t i e s - the L i n d s t r o m - R o ~ e k spaces a n d the k n o w n a s y m p t o t i c a l l y flat spaces all have Killing vectors, while the small n u m b e r o f k n o w n c o m p a c t ones are c o n s t r u c t e d a l g e b r a i c a l l y [19] - it is by no m e a n s e v i d e n t that they are sufficiently general t h a t a c o u n t e r t e r m vanishing on these spaces must be zero on all h y p e r K/ihler m a n i f o l d s . ) As so little is k n o w n a b o u t the metrics o f g e n e r a l h y p e r - K ~ h l e r spaces, it is h a r d to p r o c e e d further using the metric. H o w e v e r , we can i n s t e a d use the curvature, as all the i d e n t i t i e s satisfied by it for h y p e r - K / i h l e r spaces can be found. F o r s o m e t e n s o r R J = RE~j]kI on s o m e m a n i f o l d M to be, locally, the curvature t e n s o r for s o m e c o n n e c t i o n Fjk, it is n e c e s s a r y a n d sufficient that the Bianchi i d e n t i t y be satisfied [24]. F o r F ik to be (locally) a t o r s i o n - f r e e metric c o n n e c t i o n (i.e. the usual Christoffel c o n n e c t i o n ) for some metric g~j, it is n e c e s s a r y a n d sufficient that [24] R[ijk] I = 0 ,

Rij~kt) = 0

(A.1)

with Rokl =--Rokmg,,~. F o r that metric go to be, locally, h y p e r - K i i h l e r , it is necessary a n d sufficient that the h o l o n o m y g r o u p be S p ( m ) . T h e r e may, o f course, be t o p o l o g i cal o b s t r u c t i o n s to the g l o b a l existence o f a h y p e r - K / i h l e r metric on M - for e x a m p l e , it is n e c e s s a r y that the first C h e r n class vanishes for M to a d m i t a Ricci-flat KShler metric [18, 19] - b u t here we will only use the local p r o p e r t i e s o f the geometry. The c o n s e q u e n c e s o f the h o l o n o m y g r o u p c o n d i t i o n are best seen in a certain o r t h o n o r m a l frame. T h e n the scalar J,,L is c o n s t r u c t e d from the f r a m e c o m p o n e n t s o f the curvature a n d its c o v a r i a n t derivatives, with all indices c o n t r a c t e d using the c o n s t a n t tangent

194

C.M. Hull / Supersymmetric non-linear o" models

space metric 6 ;f, so that the metric gu, about which little is known in general, is not needed for this discussion. Note that the metric in some region is completely determined, up to changes of coordinates, by the curvature tensor and a finite number of its covariant derivatives in that region [25]. We start with some arbitrary orthonormal frame V~J on some hyper-K/ihler M, (i,j,...= 1. . . . 4m). The complex s t r u c t u r e j(3)ty can be brought to the c o n s t a n t standard form

by a local SO(4m) tangent space rotation. Then by taking complex linear combinations of basis vectors, we can choose complex frame fields eft, e i a = ( e i " ) * (a, b , . . . , &/7, = 1 , . . . , 2m) such that in this complex basis (cf. (2.4))

j(30)a/~)=(i61b - i ; ~ )

J(3)'i]= ( J ( ~ ab

'

(n.2)

go = 2e(i"ei) b &~g, ,

(A.3)

6,b = goeje5 j ,

(A.4)

eaiei b = 8a b .

(A.5)

(A.2) is left invariant by a U(2m) subgroup of the tangent space group SO(4m). By a further local U(2m) rotation, the complex structure j ( l ) t can be brought to the constant form b ,

(A.6)

O1 -10 o~b

..Oab 8gb =

01 -10

(A.7)

The subgroup of U(2m) preserving (A.6), (A.7) is Sp(m). It is convenient to introduce Sp(m) indices, A, B , . . . = 1. . . . ,2m, by dividing the 2 m complex vectors ei~ into 4m pseudo-real vectors e A p ( p , q, . . . = 1, 2): effl = ¢~Aa (ei a + S2a 5e~6 ), e A2 = 6 A ( e i a -- n % e ~ ) .

(A.8)

These satisfy the pseudo-reality conditions (ear) * = ~AtJepqeff q ,

(A.9)

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195

so that the metric (A.3) becomes go = eiApe jBq,~ ~IABCpq

(p, q are Sp(1) indices.) The remaining complex structure j ( 2 ) = j ( 3 ) j ( l ) has constant frame components. The equation

(A.IO) then also

V fl(x)j F,= O~J~' )] f, = 0

implies that the spin-connections satisfy ^

A

t o i ~ j ( x ) ' ~ f = j ( x ) j f~toikf ,

(A.11)

so that using (A.2), (A.6), (A.7), (A.8) we have oJiAPBq = o)iAB6Pq ,

(A. 12)

i.e. the spin-connection, and hence the curvature, take values in the Lie algebra of Sp(m). Using the cyclic identity, the frame components of the curvature are R A p Bq Cr Ds ~ lI]AB('D~'pq~rs

(A.13)

IItA B C D = I[t( A B C D )

(A. 14)

where

is totally symmetric so that the Ricci tensor vanishes, as it must, using the antisymmetry of J2 AB, R A p Cr = ~'~BDsqSRAp Bq CrDs = 0 .

(A.15)

The Bianchi identity is Vp~A~B3CDE = 0 .

(A.16)

Then for a hyper-K~ihler space, a set of frames can be chosen such that the tangent space group is Sp(m) - indeed this can be used as an alternative definition of a hyper-K~ihler manifold. Sufficient (and necessary) integrability conditions that a given tensor R J = RE0]k; be locally the curvature tensor of a hyper-K~ihler metric are, then, (A.1), the Bianchi identities and that the holonomy and tangent space groups be Sp(m). Then the only local identities satisfied by the curvature tensor for all hyper-Kfihler manifolds are, in an Sp(m) frame, (A.13), (A.14), (A.16), together with equations obtained from these by covariant differentiation, etc. Now consider a scalar A constructed as a polynomial in the curvature and its covariant derivatives satisfying O~A = 0

(A.17)

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for all hyper-K~ihler manifolds. Then (A.17) must follow from (A.13), (A.14), (A.16). The scalar cannot be constant for all riemannian manifolds - for any given A constructed from curvatures and their derivatives, a counter-example can be found, e.g. a K~ihler space is determined locally by a K~ihler potential K, so that, as (A.17) is a non-linear differential equation for K, it is sufficient to choose K such that (A.17) does not hold. Then (A.17) cannot follow as a result of (A.1) and the Bianchi identities VEiRjk>, = 0 alone, as these are satisfied for all riemannian spaces and this would imply (A.17) for all such manifolds. However, the algebraic identities (A.13), (A.14) alone can only imply (A.17) if they also imply A = 0 (for example, A might contain the Ricci tensor (A.15) as a factor). If A, constructed from the curvature (A.13) and its covariant derivatives (with Sp(m) indices contracted with ~'~AB and Sp(1) indices with e pq (cf. (A.10)), does not vanish as a result of the symmetry property (A.14), then (A.17) must follow from the Bianchi identity (A.16). However, as a result of the anti-symmetrization in (A.16), (A.16) can only imply (A.17) if (A.16) also implies A = 0. Then, as required, we have that a scalar, A, constructed algebraically from the curvature and its covariant derivatives can only be constant (A.17) if it vanishes identically, A = 0.

Appendix B In this appendix, we will consider the restrictions on the counterterms implied by (4.20) for hyper-K~ihler spaces that are compact and for those that are noncompact but whose curvature falls off sufficiently rapidly in some asymptotic region. (It was thought that the only compact hyper-Khhler manifold was the fourdimensional space K3, but a new class of compact hyper-Khhler spaces has recently been found [19]). We shall use the language of differential forms. Let j~x) denote the form j(x) = !l(x) 2"0 d ~ ' a dq ~j

(B.1)

and j r the 2r-form given by the r-fold exterior product JAJA...AJ. Then choosing any one of the complex structures, J = j(3), say, J" = n,

(B.2)

where -q is the volume-form ( d - - 2 n ) whose local expression is 1

rl = d l . eq...id dq~IA...A d~P id

= g d~olA d~2A...A d~ d

(B.3)

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197

with g d e f i n e d by g --- det [g~fi] = (det [gij]) ~/2

(B.4)

(cf. (2.7)). Similarly, c o r r e s p o n d i n g to the m e t r i c (3.3) we have r(X)k ~gikJ j d ¢ i ,~ d ~ j = j ( ~ ) + A T ( X ) ,

j(.x) = 1-

T (x) = !2-ik T(L)I(X) J

k

j dq~'A d ~ j

(B.5)

a n d in a c c o r d a n c e with the choice o f c o m p l e x structure, we use the n o t a t i o n J = f 3 ) , T = T (3). T h e v o l u m e f o r m c o r r e s p o n d i n g to g is ~/= J " = ~ d ~ l a . . , dq~ d

(B.6)

with -= det ( ~ f i ) = g det (6~ t3 + AT~~) = g h ( A )

(B.7)

with h ( a ) the p o l y n o m i a l with c o n s t a n t coefficients defined in (4.20). T h e n fi=h(h)n

(B.8)

a n d so, f r o m (B.2), ( J + AT)" = h ( A ) J "

(B.9)

m u s t h o l d for all A. F r o m (3.2), we have d T (x) = 0.

(B.10)

S u p p o s e n o w that M is c o m p a c t . T h e n b y the H o d g e d e c o m p o s i t i o n t h e o r e m [18], T has a u n i q u e d e c o m p o s i t i o n T=dA

+H

(B.11)

for s o m e o n e - f o r m A a n d s o m e h a r m o n i c t w o - f o r m H, A H = 0 with A the H o d g e - d e R h a m l a p l a c i a n (A = d6 + 6d, where 6 is the d i v e r g e n c e o p e r a t o r that is a d j o i n t to d ; on p - f o r m s 6 = ( - 1 ) P * d *, with * the d u a l i t y o p e r a t o r m a p p i n g p - f o r m s to ( d - p ) - f o r m s [18]). As the Kfihler form J is h a r m o n i c for c o m p a c t K/ihler spaces, it follows that the r i g h t - h a n d side o f (B.9) is h a r m o n i c , w h e r e a s from (B.10) the l e f t - h a n d side is the sum o f a h a r m o n i c form ( J + A H ) " a n d a c l o s e d form, a n d can only be h a r m o n i c i f A = 0. Then, from (B.11), T = T (3) must be a h a r m o n i c t w o - f o r m , A T = O . By r e p e a t i n g this a r g u m e n t with the o t h e r choices o f c o m p l e x structure, j = j(l) a n d J j(2), we find that T I~) a n d T 12) m u s t also be h a r m o n i c . The h a r m o n i c forms T (x) c a n n o t be l i n e a r l y d e p e n d e n t , since if aT¢~)+ bT(2)+ c T (3) = 0, it follows that aJ(l)+ b J(2)+ cJ (3) = 0. S q u a r i n g this gives a 2 + b e + c 2 = 0 so that a = b = c = 0. =

The n u m b e r o f l i n e a r l y i n d e p e n d e n t h a r m o n i c p - f o r m s is a t o p o l o g i c a l invariant, the p t h Betti n u m b e r bp.

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198

On any Kiihler manifold, a (p, q)-form is one which, in complex coordinates, has p holomorphic indices and q anti-holomorphic ones o)

1

.

P ! q ! ° 2 ~ ' ..~d~...t~,dz ,A

dz ~ A d : ~ ' A . . . d ~ "

In complex coordinates adapted to j(3), the K5hler form j(3) and T (3) are of type (1, 1) while J~), j~2) are given in terms of the (2, 0)-form g2 = ½g2~o dZ~A dz ° (with O~o = g~aS2"o as defined in (4.16)) and the complex conjugate (0, 2)-form O by j~l) = ½(O + ~0) '

j(2)= _ l i ( . O _ g ~ ) .

(B.12a)

T(2) = ½ i ( o ' - # ) .

(B.12b)

There is also a (2, 0)-form o" such that T ~ ) = ½(o-5- ~ ) ,

The number of independent harmonic (p, q)-forms, bp, q, (bp, q = bq,v) is a topological invariant so that this greatly restricts the possible forms of T. For example, for K3, bl,~ = 20 and b2,o = bo,2 = 1. Then there is only one independent harmonic (2, 0)-form, so that from (B.12) cr = 2ag2 for some complex constant a, and T (1) = a/2 + a*,0 = (a + a * ) j ( 1 ) q - i(a - a * ) J (2) .

(B.13)

From (B.5), this gives --oT(L)a- ( a- + a*) u gu- + i(

~,~.c3) ~a o

(B.14)

As T ~ ) is symmetric, the constant a must be real and we have that -T(L) 0 -= 2ago. Then if either b~,~ or b2.o are unity, (as is the case for K3) the renormalized metric must be proportional to the original metric. Such counterterms are ruled out by the arguments of sect. 4 and appendix A, as the constant a would have to be constructed from curvatures and their covariant derivatives in order to scale correctly and such constant curvature scalars do not exist. More generally, the counterterms -T0(L) a l l lie in a finite dimensional linear space of dimension not greater than either b~.~ or b2,0 (which are both positive as j(x) are harmonic). Consider now the case of non-compact M and suppose that the closed form q(B.10) is exact, T = dA, for some one-form A. Then ( J + AT)" = J" + d B

(B.15)

for a ( d - 1)-form B = A A A J n l - t - ' ' ' + A " A A T " i. Let Me be a sequence of finitevolume d-dimensional subspaces of M with boundary 0Mn, such that as R ~ oo, MR ~ M. (We shall assume that there is such a sequence). Then integrating (B.15) asing (B.2), (B.6), (B.8), we obtain h(A)VR=f

h(A)r/=fM MR

~=f R

(-o+dB)=VR+f~, MR

B, MR

(B.16)

C.M. hull / Supersymmetric non-linear o- models

199

where the v o l u m e o f MR is

VR= f

ft.

(B.17)

MR

Then h ( h ) = 1 +-~--~R

B.

(B.18)

MR

The l e f t - h a n d side is a c o n s t a n t , i n d e p e n d e n t o f R. S u p p o s e the c o m p o n e n t s o f the c u r v a t u r e tensor, a n d h e n c e B, which is c o n s t r u c t e d from c u r v a t u r e s a n d their c o v a r i a n t derivatives, fall off sufficiently fast that lim

,f

R~oc ~ R

B = 0.

(B. 19)

,~MR

T h e n h ( h ) = 1 a n d so, by the a r g u m e n t s o f sect. 4, T m u s t vanish. Thus, for - - i!~) 3 m a n i f o l d s w h o s e b o u n d a r y c o n d i t i o n s are such that (B.19) holds, if the form T is exact it m u s t vanish, so that this rules out all forms T = T (3) e x c e p t those that are c l o s e d b u t n o t exact, i.e. t h e y m u s t be in s o m e n o n - t r i v i a l c o h o m o l o g y class. This greatly limits the p o s s i b l e c o u n t e r t e r m s . As an e x a m p l e , c o n s i d e r spaces w h o s e m e t r i c tends a s y m p t o t i c a l l y to that o f flat space, at least locally, gij =

t~ijq-O(r-a)

as r ~ o o

(B.20)

with a s o m e positive c o n s t a n t a n d r s o m e r a d i a l c o o r d i n a t e such that the a r e a o f a ( d - 1 ) - d i m e n s i o n a l " s p h e r e " r = R scales as R d-1 (we will take the i n t e r i o r of r = R to be MR), while the c o m p o n e n t s o f the c u r v a t u r e a n d its c o v a r i a n t derivatives in some o r t h o - n o r m a l f r a m e fall off as

R H d = O ( r -b) V~, ... V ~pR ~ f = O ( r -b-pc)

as r ~ cx3, as r ~ o o

(B.21) (B.22)

for s o m e n o n - n e g a t i v e constants b, c. S u p p o s e that T ~ ) is c o n s t r u c t e d from C i c u r v a t u r e t e n s o r s Rjkl, 2 D c o v a r i a n t derivatives Vi, E inverse metrics g'J" a n d F metrics go (with i n d i c e s as i n d i c a t e d here). T h e n as Vi a n d R~kl d o not scale u n d e r (4.21), it f o l l o w s from (4.21), (4.22) that E - F = L - 1, while the r e q u i r e m e n t that the n u m b e r o f c o v a r i a n t i n d i c e s exceed the n u m b e r o f c o n t r a v a r i a n t i n d i c e s b y two gives

C + D = E-F+

1 = L.

(B.23)

as R ~ oo

(B.24)

T h e n as B = AAA Jn ~+ . . • it follows that

fo

B = O ( R x) MR

200

C.M. H u l l / Supersymmetric non-linear o, models

with c~MR the ( d - 1)-surface r = R and x~

d-

l-Cb-(2D-1)c=

= d - 1 + c - 2cL-

d-

l + c-

Lb-

2c).

C(b -

D(2c-b)

(B.25)

With these b o u n d a r y conditions, V R - - O ( R d) as R->oo, so that we will have the required scaling behaviour (B.19) if x < d. If 2 c ~ b, as L ~ 1, x- d ~ c(l-2L)-

l~-(c+l)

<0,

(B.26)

so that (B.19) holds automatically, while if b < 2c, x - d < c - 1 - Lb,

(B.27)

so that (B.19) holds for all loop orders L if c ~ 1 , while if c > 1, (B.19) holds for all loop orders b e y o n d the constant lower b o u n d C--1

L>--

b

(B.28)

However, for hyper-K~ihler spaces, T ~L) = 0 for L < 4 [5, 6] so that if ~b< c < 4b + 1, (B.19) will again hold to all orders. Note that the frame c o m p o n e n t s o f -T- t j!5) fall off as T(L) /j = O ( r (bC+2cD))

(B.29)

For any space with (B.19) satisfied, the T (x) must be closed but not exact and we shall suppose that it vanishes asymptotically, as in (B.29). The n u m b e r o f linearly i n d e p e n d e n t p-forms ((p, q)-forms) on M that are closed but not exact which vanish asymptotically is a topological invariant, the Betti n u m b e r B v ( B p , q), which is also the dimension of the (relative) c o h o m o l o g y g r o u p HV(M)(HP'q(M)). Again T ~3) is a (1, l ) - f o r m while T (~), T (2) are o f type (2, 0) + (0, 2). Then T(,~ ) can only be non-zero if B2,0= Bo,2~ 1 and B~.I ~ 1, so that if B2 = B ~ a + 2 B 2 , 0 < 3 , the c o r r e s p o n d i n g nonlinear sigma model is finite, while if Bla = 1 or B2.o= 1 the renormalized metric tensor must be proportional to the original metric, ( N o t e that for n o n - c o m p a c t hyper-K~ihler spaces, J(~) can be exact, so that B~.~ and B2,o can vanish). For example, consider the case d = 4 in which M is an (anti-)self-dual gravitational instanton. The m u l t i - T a u b - N U T and m u l t i - E g u c h i - H a n s e n metrics [26] satisfy the b o u n d a r y conditions considered above, so that (B.19) and (B.29) hold. The self-dual T a u b - N U T space is topologically equivalent to R 4 [26] so that B 2 = 0 and the c o r r e s p o n d i n g sigma-model is finite. The m u l t i - T a u b - N U T metrics with s " s o u r c e s " or nuts [26] have B 2 - - s - 1 so that for s ~ 3 these models are necessarily finite to all orders. The E g u c h i - H a n s e n space has B2 = 1, while its s-nut generalization has B2 = s - 1, so that we again have finiteness for s ~ 3. We have seen that, at least when M satisfies certain b o u n d a r y conditions, the q u a n t u m corrections to the metric have a topological character, forcing the -To!~ to

C.M. Hull / Supersymmetric non-linear o" models

201

lie in s o m e finite d i m e n s i o n a l linear space for all l o o p orders, placing a great restriction on the counterterms. For s o m e t o p o l o g i e s all c o u n t e r t e r m s must vanish, while for others, T _!j(L) m u s t be proportional to the metric. Typically, the few tensors T~j that are a l l o w e d by these considerations do not scale correctly to occur in perturbation theory [7], so that, by the arguments of sect. 4, -T- O!~) must be o f the form _,iT !~) = A T o where A is a constant scalar constructed from curvatures and their covariant derivatives in such a way that the counterterm scales correctly. The arguments o f appendix A are then n e e d e d to s h o w that there are no such nonvanishing scalars so as to rule out this possibility, but as was seen in sect. 4, these arguments give the finiteness result directly. Note added in proof There are further papers on the finiteness o f the N = 4 m o d e l s by M u c k and by Galperin, Ivanov, Ogievetsky and Sokatchev, "Proofs" of the finiteness of the N = 1 Ricci-flat m o d e l s by m y s e l f [IAS preprint] and by A l v a r e z - G a u m ~ and Ginsparg [Harvard preprint] turned out to be incorrect. In a recent paper, A l v a r e z - G a u m 6 , C o l e m a n and Ginsparg argue that the N = 2 Ricci-flat m o d e l s are finite.

References [1] D. Friedan, Phys. Rev. Lett. 45 (1980) 1057 [2] D.Z. Freedman and P.K. Towflsend, Nucl. Phys. B177 (1981) 282 [3] M. Ro~ek, Supersymmetry and non-linear o--models, Proc. of the Supersymmetry in physics conference, Los Alamos, Dec. 1983, to be published [4] L. Alvarez-Gaum~ and D.Z. Freedman, Phys. Rev. D15 (1980) 846 [5] L. Alvarez-Gaum~, D.Z. Freedman and S. Mukhi, Ann. Phys. 134 (1981) 85 [6] L. Alvarez-Gaum~, Nucl. Phys. B184 (1981) 180 [7] L. Alvarez-Gaum6 and D.Z. Freedman, C o m m u n . Math. Phys. 80 (1981) 443 [8] J.H. Schwarz, Phys. Rep. 89 (1982) 223; M.B. Green, Surveys in high energy physics, 3 (1983) 127 [9] D. Friedan, Z. Qiu and S. Shenker, to appear: D. J. Gross and E. Witten, to appear: P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for'superstrings, 1TP-UCSB preprint (1984) [10] B. Zumino, Phys. Lett. 87B (1979) 203 [11] L. Alvarez-Gaum~ and D.Z. Freedman, in Unification of the fundamental particle interactions, eds S. Ferrara, J. Ellis and P. van Nieuwenhuizen, Plenum Press, N.Y. (1980) [12] A. Lichnerowicz, Th6orie globale des connections et des groupes d'holonomie, Consiglio Nazionale delle Richerche (1955) [13] S.S. Chern, Complex manifolds without potential theory (Van Nostrand, Princeton, 1969) [14] S.J. Gates, Jr., C.M. Hull and M. Ro~ek, Nucl. Phys. B248 (1984) 157 [15] L. Alvarez-Gaum~ and D.Z. Freedman, C o m m u n . Math. Phys. 91 (1983) 87, S.J. Gates, Jr., Nucl. Phys. B238 (1984) 349 [16] W. Siegel, Phys. Lett. 84B (1979) 193 [17] B. de Wit and M.T. Grisaru, Phys. Rev. D20 (1979) 2082 [18] T. Aubin, Nonlinear analysis on manifolds. Monge-Amp~re equations, (Springer, New York, 1982) [19] A. Beauville, J. Diff. Geom. 18 (1983) 755

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C.M. Hull / Supersymmetric non-linear o" models

[20] C.M. Hull, in preparation [21] T.L. Curtright and C. K. Zachos, Phys. Rev. Lett. 53 (1984) 1799; R. Rohm, Princeton preprint (1984) [22] U. Lindstrom and M. Ro~ek, Nucl. Phys. B222 (1983) 285 [23] E. Calabi, Ann. Sci. de I'E.N.S. 12 (1979) 266; L. Alvarez-Gaum6 and D.Z. Freedman, Phys. Lett. 94B (1980) 171 [24] J.A. Schouten, Ricci calculus (Springer, Berlin, 1954) [25] E. Cartan, Leqons sur la g6ometrie des espaces de Riemann (Gauthier-Villars Paris, 1946) A. Karlhede, Gen. Rel. Gray. 12 (1980) 693 [26] G.W. Gibbons and S.W. Hawking, Commun. Math. Phys. 66 (1979) 291